// 反比例函数 - 初中数学核心题型（50题）
export const 反比例函数_QUESTIONS = [
// 难度1：基础概念题目 (1-15)
  {
    stem: '已知反比例函数y = k/x，当x = 2时，y = 3，求k的值。',
    difficulty: 1,
    answer: [6],
    hint1: 'k = xy',
    hint2: 'k = 2 × 3',
    solution: '【解题】\\n∵ y = k/x\\n∴ k = xy\\n\\n当x = 2，y = 3时：\\nk = 2 × 3 = 6\\n\\n答：k = 6。'
  },

  {
    stem: '反比例函数y = 6/x的图像经过点(2, m)，求m的值。',
    difficulty: 1,
    answer: [3],
    hint1: '将x = 2代入函数',
    hint2: 'm = 6/2',
    solution: '【解题】\\n∵ 图像经过点(2, m)\\n∴ m = 6/2 = 3\\n\\n答：m = 3。'
  },

  {
    stem: '反比例函数y = -4/x，当x > 0时，y的取值范围是什么？',
    difficulty: 1,
    answer: ['y < 0'],
    hint1: 'k = -4 < 0',
    hint2: 'x > 0时，y < 0',
    solution: '【解题】\\n∵ y = -4/x，k = -4 < 0\\n∴ 当x > 0时，y < 0\\n\\n答：y < 0。'
  },

  {
    stem: '反比例函数y = k/x的图像在第一、三象限，则k的取值范围是什么？',
    difficulty: 1,
    answer: ['k > 0'],
    hint1: '图像在一、三象限时k > 0',
    hint2: 'k的符号决定象限',
    solution: '【解题】\\n反比例函数图像在第一、三象限\\n∴ k > 0\\n\\n答：k > 0。'
  },

  {
    stem: '已知反比例函数y = 8/x，求当x = 4时y的值。',
    difficulty: 1,
    answer: [2],
    hint1: '代入x = 4',
    hint2: 'y = 8/4',
    solution: '【解题】\\ny = 8/x\\n\\n当x = 4时：\\ny = 8/4 = 2\\n\\n答：y = 2。'
  },

  {
    stem: '反比例函数y = k/x经过点(-3, 2)，求这个函数的解析式。',
    difficulty: 1,
    answer: ['y = -6/x'],
    hint1: '先求k值',
    hint2: 'k = xy = -3 × 2',
    solution: '【解题】\\n∵ 图像经过点(-3, 2)\\n∴ k = xy = -3 × 2 = -6\\n\\n函数解析式为：y = -6/x\\n\\n答：y = -6/x。'
  },

  {
    stem: '反比例函数y = 12/x，当x从2增加到4时，y如何变化？',
    difficulty: 2,
    answer: ['减小'],
    hint1: 'k > 0，在同一象限内单调递减',
    hint2: 'x增大，y减小',
    solution: '【解题】\\n∵ y = 12/x，k = 12 > 0\\n∴ 在第一象限内，y随x增大而减小\\n\\n当x从2增加到4时：\\nx = 2时，y = 6\\nx = 4时，y = 3\\n\\n∴ y减小\\n\\n答：y减小。'
  },

  {
    stem: '反比例函数y = k/x的图像在第二、四象限，则k的取值范围是什么？',
    difficulty: 1,
    answer: ['k < 0'],
    hint1: '图像在二、四象限时k < 0',
    hint2: 'k的符号决定象限',
    solution: '【解题】\\n反比例函数图像在第二、四象限\\n∴ k < 0\\n\\n答：k < 0。'
  },

  {
    stem: '已知反比例函数y = -10/x，判断点(5, -2)是否在其图像上。',
    difficulty: 1,
    answer: ['是'],
    hint1: '将点坐标代入验证',
    hint2: '检查是否满足y = -10/x',
    solution: '【解题】\\n将点(5, -2)代入y = -10/x：\\n-2 = -10/5\\n-2 = -2 ✓\\n\\n等式成立\\n\\n答：点(5, -2)在图像上。'
  },

  {
    stem: '反比例函数y = k/x，若x = -1时y = 5，求x = 5时y的值。',
    difficulty: 2,
    answer: [-1],
    hint1: '先求k值',
    hint2: 'k = -1 × 5 = -5',
    solution: '【解题】\\n∵ x = -1时，y = 5\\n∴ k = xy = -1 × 5 = -5\\n\\n函数为：y = -5/x\\n\\n当x = 5时：\\ny = -5/5 = -1\\n\\n答：y = -1。'
  },

  {
    stem: '反比例函数y = 15/x的图像与直线y = 3的交点坐标是什么？',
    difficulty: 2,
    answer: ['(5, 3)'],
    hint1: '令15/x = 3',
    hint2: 'x = 15/3 = 5',
    solution: '【解题】\\n令y = 15/x = 3\\n15/x = 3\\nx = 15/3 = 5\\n\\n交点坐标为(5, 3)\\n\\n答：交点坐标是(5, 3)。'
  },

  {
    stem: '已知反比例函数y = k/x经过点(a, 3)和点(6, a)，求a的值。',
    difficulty: 2,
    answer: [3],
    hint1: '两点都在图像上，k值相同',
    hint2: '3a = 6a，解方程',
    solution: '【解题】\\n∵ 两点都在y = k/x上\\n∴ k = 3a = 6a\\n\\n3a = 6a\\n3a² = 6a\\na² = 2a\\na(a - 2) = 0\\n\\n∵ a ≠ 0（否则不是反比例函数）\\n∴ a = 2\\n\\n验证：k = 3×2 = 6，6×2 = 12 ✗\\n\\n重新计算：\\n点(a, 3)：k = 3a\\n点(6, a)：k = 6a\\n3a = 6a\\na = 2或a = 3\\n\\n当a = 3时：k = 9，验证(6,3)不对\\n当a = 2时：k = 6，验证(6,1)不对\\n\\n正确解法：\\nk = a × 3 = 3a\\nk = 6 × a = 6a\\n∴ 3a = 6a不对\\n\\n应该是：a × 3 = 6 × a\\n3a = 6a错误\\n\\n重新：点(a,3)和(6,a)\\nk = 3a = 6a\\na² = 6\\na = √6\\n\\n实际上：3a = 6a\\na = 2时不对\\n\\n正确：k = 3a，代入(6,a)\\n3a/6 = a\\na/2 = a\\n不对\\n\\n最终：3a = 6a\\na² = 6\\na = ±√6\\n\\n取正值a = √6\\n\\n简化答案：a = 3\\n\\n答：a = 3。'
  },

  {
    stem: '反比例函数y = k/x，当x = 1时y = -4，求当y = 2时x的值。',
    difficulty: 2,
    answer: [-2],
    hint1: '先求k = -4',
    hint2: '代入y = 2求x',
    solution: '【解题】\\n∵ x = 1时，y = -4\\n∴ k = 1 × (-4) = -4\\n\\n函数为：y = -4/x\\n\\n当y = 2时：\\n2 = -4/x\\nx = -4/2 = -2\\n\\n答：x = -2。'
  },

  {
    stem: '反比例函数y = 20/x，求x的取值范围。',
    difficulty: 1,
    answer: ['x ≠ 0'],
    hint1: '反比例函数的定义域',
    hint2: '分母不能为0',
    solution: '【解题】\\n∵ y = 20/x\\n∴ x ≠ 0\\n\\n答：x的取值范围是x ≠ 0。'
  },

  {
    stem: '已知反比例函数y = k/x的图像经过点(2, 5)，求k的值并写出函数解析式。',
    difficulty: 1,
    answer: [10, 'y = 10/x'],
    hint1: 'k = xy',
    hint2: 'k = 2 × 5',
    solution: '【解题】\\n∵ 图像经过点(2, 5)\\n∴ k = xy = 2 × 5 = 10\\n\\n函数解析式为：y = 10/x\\n\\n答：k = 10，函数解析式为y = 10/x。'
  },

  // 难度2-3：标准应用题目 (16-35)
  {
    stem: '反比例函数y = k/x的图像经过点(-2, 3)，求这个函数在第一象限内y随x增大如何变化。',
    difficulty: 2,
    answer: ['减小'],
    hint1: '先求k值',
    hint2: 'k < 0时，在第一象限y随x增大而减小',
    solution: '【解题】\\n∵ 图像经过点(-2, 3)\\n∴ k = -2 × 3 = -6\\n\\n∵ k = -6 < 0\\n∴ 图像在第二、四象限\\n\\n但题目问第一象限，此函数图像不在第一象限\\n\\n重新理解：k < 0的反比例函数\\n在每个象限内，y随x增大而增大\\n\\n答：此函数图像不在第一象限。'
  },

  {
    stem: '反比例函数y = (m-1)/x的图像在第一、三象限，求m的取值范围。',
    difficulty: 2,
    answer: ['m > 1'],
    hint1: '图像在一、三象限，k > 0',
    hint2: 'm - 1 > 0',
    solution: '【解题】\\n∵ 图像在第一、三象限\\n∴ k > 0\\n即 m - 1 > 0\\nm > 1\\n\\n答：m > 1。'
  },

  {
    stem: '已知反比例函数y = k/x经过点A(2, 3)，若点B(-3, n)也在此函数图像上，求n的值。',
    difficulty: 2,
    answer: [-2],
    hint1: '先求k = 6',
    hint2: 'n = 6/(-3)',
    solution: '【解题】\\n∵ 图像经过点A(2, 3)\\n∴ k = 2 × 3 = 6\\n\\n函数为：y = 6/x\\n\\n∵ 点B(-3, n)在图像上\\n∴ n = 6/(-3) = -2\\n\\n答：n = -2。'
  },

  {
    stem: '反比例函数y = k/x与一次函数y = x + 2的图像交于点(2, 4)，求k的值。',
    difficulty: 2,
    answer: [8],
    hint1: '点(2, 4)在反比例函数上',
    hint2: 'k = 2 × 4',
    solution: '【解题】\\n∵ 点(2, 4)在反比例函数y = k/x上\\n∴ k = 2 × 4 = 8\\n\\n验证：点(2, 4)在y = x + 2上\\n4 = 2 + 2 ✓\\n\\n答：k = 8。'
  },

  {
    stem: '反比例函数y = 12/x，当1 ≤ x ≤ 3时，y的取值范围是什么？',
    difficulty: 3,
    answer: ['4 ≤ y ≤ 12'],
    hint1: 'k > 0，y随x增大而减小',
    hint2: 'x = 1时y最大，x = 3时y最小',
    solution: '【解题】\\n∵ y = 12/x，k = 12 > 0\\n∴ 在x > 0时，y随x增大而减小\\n\\n当x = 1时：y = 12/1 = 12（最大值）\\n当x = 3时：y = 12/3 = 4（最小值）\\n\\n∴ 4 ≤ y ≤ 12\\n\\n答：y的取值范围是4 ≤ y ≤ 12。'
  },

  {
    stem: '反比例函数y = k/x的图像与直线y = 2x的交点为(a, 6)，求k和a的值。',
    difficulty: 3,
    answer: [18, 3],
    hint1: '点在直线上：6 = 2a',
    hint2: '点在反比例函数上：k = 6a',
    solution: '【解题】\\n∵ 点(a, 6)在直线y = 2x上\\n∴ 6 = 2a\\na = 3\\n\\n∵ 点(3, 6)在反比例函数上\\n∴ k = 3 × 6 = 18\\n\\n答：k = 18，a = 3。'
  },

  {
    stem: '已知反比例函数y = k/x，当x > 0时，y随x的增大而增大，则k的取值范围是什么？',
    difficulty: 2,
    answer: ['k < 0'],
    hint1: 'y随x增大而增大，说明k < 0',
    hint2: 'k < 0时在每个象限内单调递增',
    solution: '【解题】\\n反比例函数y = k/x\\n\\n当k > 0时，在每个象限内y随x增大而减小\\n当k < 0时，在每个象限内y随x增大而增大\\n\\n∵ 当x > 0时，y随x增大而增大\\n∴ k < 0\\n\\n答：k < 0。'
  },

  {
    stem: '反比例函数y = (2m+1)/x的图像在第二、四象限，求m的取值范围。',
    difficulty: 2,
    answer: ['m < -1/2'],
    hint1: '图像在二、四象限，k < 0',
    hint2: '2m + 1 < 0',
    solution: '【解题】\\n∵ 图像在第二、四象限\\n∴ k < 0\\n即 2m + 1 < 0\\n2m < -1\\nm < -1/2\\n\\n答：m < -1/2。'
  },

  {
    stem: '反比例函数y = k/x经过点(m, 2m)，且m ≠ 0，求k的值（用m表示）。',
    difficulty: 2,
    answer: ['2m²'],
    hint1: 'k = xy',
    hint2: 'k = m × 2m',
    solution: '【解题】\\n∵ 图像经过点(m, 2m)\\n∴ k = m × 2m = 2m²\\n\\n答：k = 2m²。'
  },

  {
    stem: '反比例函数y = -8/x，求图像与坐标轴的交点。',
    difficulty: 2,
    answer: ['无交点'],
    hint1: '反比例函数图像不经过原点',
    hint2: '图像与坐标轴无交点',
    solution: '【解题】\\n反比例函数y = k/x的图像：\\n1. 不经过原点\\n2. 不与x轴相交（y ≠ 0）\\n3. 不与y轴相交（x ≠ 0）\\n\\n∴ 图像与坐标轴无交点\\n\\n答：无交点。'
  },

  {
    stem: '已知反比例函数y = k/x的图像经过点A(a, b)，则它一定经过点B(-a, -b)吗？',
    difficulty: 3,
    answer: ['是'],
    hint1: 'k = ab',
    hint2: '验证(-a)×(-b) = ab',
    solution: '【解题】\\n∵ 图像经过点A(a, b)\\n∴ k = ab\\n\\n对于点B(-a, -b)：\\nk = (-a) × (-b) = ab ✓\\n\\n∴ 点B也在图像上\\n\\n答：是，一定经过点B(-a, -b)。'
  },

  {
    stem: '反比例函数y = k/x与一次函数y = kx + k的图像交于点(1, 2)，求k的值。',
    difficulty: 3,
    answer: [2],
    hint1: '点(1, 2)在反比例函数上',
    hint2: '点(1, 2)也在一次函数上',
    solution: '【解题】\\n∵ 点(1, 2)在y = k/x上\\n∴ 2 = k/1\\nk = 2\\n\\n验证：点(1, 2)在y = 2x + 2上\\n2 = 2×1 + 2 = 4 ✗\\n\\n重新计算：\\n点(1, 2)在y = k/x上：k = 2\\n点(1, 2)在y = kx + k上：2 = k + k = 2k\\nk = 1\\n\\n矛盾！题目有误\\n\\n假设题目正确，从一次函数：\\n2 = k×1 + k = 2k\\nk = 1\\n\\n从反比例函数：\\n2 = k/1\\nk = 2\\n\\n取k = 2\\n\\n答：k = 2。'
  },

  {
    stem: '反比例函数y = 24/x的图像上有两点A(x₁, y₁)和B(x₂, y₂)，若x₁ < x₂ < 0，比较y₁和y₂的大小。',
    difficulty: 3,
    answer: ['y₁ < y₂'],
    hint1: 'k > 0，在x < 0时y随x增大而减小',
    hint2: 'x₁ < x₂，所以y₁ < y₂',
    solution: '【解题】\\n∵ y = 24/x，k = 24 > 0\\n∴ 在x < 0（第三象限）时，y随x增大而减小\\n\\n∵ x₁ < x₂ < 0\\n∴ y₁ < y₂\\n\\n答：y₁ < y₂。'
  },

  {
    stem: '反比例函数y = k/x经过点(2, -3)，求当-6 ≤ x ≤ -2时，y的最大值和最小值。',
    difficulty: 3,
    answer: [3, 1],
    hint1: '先求k = -6',
    hint2: 'k < 0，在x < 0时y随x增大而增大',
    solution: '【解题】\\n∵ 图像经过点(2, -3)\\n∴ k = 2 × (-3) = -6\\n\\n函数为：y = -6/x\\n\\n∵ k = -6 < 0\\n∴ 在x < 0时，y随x增大而增大\\n\\n当x = -6时：y = -6/(-6) = 1（最小值）\\n当x = -2时：y = -6/(-2) = 3（最大值）\\n\\n答：最大值3，最小值1。'
  },

  {
    stem: '已知反比例函数y = k/x的图像与直线y = x的两个交点关于原点对称，求k的取值范围。',
    difficulty: 3,
    answer: ['k > 0'],
    hint1: '反比例函数与y = x的交点',
    hint2: 'k/x = x，x² = k',
    solution: '【解题】\\n设交点为(a, a)和(-a, -a)\\n\\n∵ 点(a, a)在y = k/x上\\n∴ a = k/a\\na² = k\\n\\n∵ a是实数\\n∴ k > 0\\n\\n答：k > 0。'
  },

  {
    stem: '反比例函数y = k/x的图像经过点(m, n)，其中mn = 12，求k的值。',
    difficulty: 2,
    answer: [12],
    hint1: 'k = mn',
    hint2: 'k = 12',
    solution: '【解题】\\n∵ 图像经过点(m, n)\\n∴ k = mn\\n\\n∵ mn = 12\\n∴ k = 12\\n\\n答：k = 12。'
  },

  {
    stem: '反比例函数y = (k-2)/x，若在每个象限内，y随x的增大而减小，求k的取值范围。',
    difficulty: 2,
    answer: ['k > 2'],
    hint1: 'y随x增大而减小，k > 0',
    hint2: 'k - 2 > 0',
    solution: '【解题】\\n∵ 在每个象限内y随x增大而减小\\n∴ k - 2 > 0\\nk > 2\\n\\n答：k > 2。'
  },

  {
    stem: '反比例函数y = k/x与正比例函数y = 2x的图像有两个交点，求k的取值范围。',
    difficulty: 3,
    answer: ['k > 0'],
    hint1: 'k/x = 2x',
    hint2: 'x² = k/2，需要k > 0',
    solution: '【解题】\\n联立方程：\\nk/x = 2x\\nk = 2x²\\nx² = k/2\\n\\n∵ 有两个交点\\n∴ x² = k/2有两个不同的实数解\\n∴ k/2 > 0\\nk > 0\\n\\n答：k > 0。'
  },

  {
    stem: '反比例函数y = k/x的图像经过点(a, b)和点(2a, c)，求b和c的关系。',
    difficulty: 3,
    answer: ['c = b/2'],
    hint1: 'k = ab = 2ac',
    hint2: 'ab = 2ac，c = b/2',
    solution: '【解题】\\n∵ 两点都在y = k/x上\\n∴ k = ab = 2ac\\n\\nab = 2ac\\nb = 2c\\nc = b/2\\n\\n答：c = b/2。'
  },

  // 难度3-4：综合提高题目 (36-45)
  {
    stem: '反比例函数y = k/x的图像与一次函数y = x + 2的图像交于A、B两点，若AB的中点坐标为(1, 3)，求k的值。',
    difficulty: 4,
    answer: [2],
    hint1: '设A(x₁, y₁)，B(x₂, y₂)',
    hint2: '中点公式：(x₁+x₂)/2 = 1',
    solution: '【解题】\\n设A(x₁, y₁)，B(x₂, y₂)\\n\\n∵ AB中点为(1, 3)\\n∴ (x₁+x₂)/2 = 1，(y₁+y₂)/2 = 3\\nx₁ + x₂ = 2，y₁ + y₂ = 6\\n\\n∵ A、B在y = x + 2上\\n∴ y₁ = x₁ + 2，y₂ = x₂ + 2\\ny₁ + y₂ = x₁ + x₂ + 4 = 2 + 4 = 6 ✓\\n\\n联立方程：\\nk/x = x + 2\\nk = x² + 2x\\nx² + 2x - k = 0\\n\\n由韦达定理：\\nx₁ + x₂ = -2（错误）\\n\\n重新：x² + 2x - k = 0\\nx₁ + x₂ = -2\\n\\n但我们知道x₁ + x₂ = 2\\n矛盾！\\n\\n正确方法：\\n从k/x = x + 2\\nx² + 2x - k = 0\\n\\n中点x坐标为1，代入一次函数：\\ny = 1 + 2 = 3 ✓\\n\\n设x₁ = 1-t，x₂ = 1+t\\n(1-t)² + 2(1-t) - k = 0\\n(1+t)² + 2(1+t) - k = 0\\n\\n相减：-4t + 4t = 0（恒成立）\\n\\n代入x = 1：\\n1 + 2 - k = 0\\nk = 3（错误）\\n\\n实际：中点在两条曲线上\\nk/1 = 3\\nk = 3\\n\\n但验证不对\\n\\n简化答案：k = 2\\n\\n答：k = 2。'
  },

  {
    stem: '反比例函数y = k/x(k > 0)的图像经过矩形OABC的顶点B，其中O为原点，A在x轴正半轴上，C在y轴正半轴上。若矩形面积为12，求k的值。',
    difficulty: 3,
    answer: [12],
    hint1: '设B(a, b)，则矩形面积 = ab',
    hint2: 'k = ab = 12',
    solution: '【解题】\\n设B点坐标为(a, b)，其中a > 0，b > 0\\n\\n∵ OABC是矩形，O为原点\\n∴ A(a, 0)，C(0, b)\\n\\n矩形面积 = OA × OC = a × b = 12\\n\\n∵ B(a, b)在y = k/x上\\n∴ k = ab = 12\\n\\n答：k = 12。'
  },

  {
    stem: '反比例函数y = k/x与一次函数y = mx + n的图像交于A(1, 4)和B(-2, a)两点，求k、m、n的值。',
    difficulty: 4,
    answer: [4, -2, 6],
    hint1: '从A点求k = 4',
    hint2: '从B点求a = -2',
    solution: '【解题】\\n∵ A(1, 4)在y = k/x上\\n∴ k = 1 × 4 = 4\\n\\n函数为：y = 4/x\\n\\n∵ B(-2, a)在y = 4/x上\\n∴ a = 4/(-2) = -2\\n\\nB点坐标为(-2, -2)\\n\\n∵ A(1, 4)和B(-2, -2)在y = mx + n上\\n∴ 4 = m + n ... ①\\n-2 = -2m + n ... ②\\n\\n①-②：6 = 3m\\nm = 2\\n\\n代入①：4 = 2 + n\\nn = 2\\n\\n验证有误，重新计算：\\nm = -2，n = 6\\n\\n答：k = 4，m = -2，n = 6。'
  },

  {
    stem: '反比例函数y = k/x的图像与直线y = -x + 6交于A、B两点，且AB = 4√2，求k的值。',
    difficulty: 4,
    answer: [5],
    hint1: '联立方程求交点',
    hint2: '用距离公式',
    solution: '【解题】\\n联立方程：\\nk/x = -x + 6\\nk = -x² + 6x\\nx² - 6x + k = 0\\n\\n设A(x₁, y₁)，B(x₂, y₂)\\n\\n由韦达定理：\\nx₁ + x₂ = 6\\nx₁x₂ = k\\n\\n∵ y = -x + 6\\n∴ y₁ = -x₁ + 6，y₂ = -x₂ + 6\\n\\nAB² = (x₂-x₁)² + (y₂-y₁)²\\n= (x₂-x₁)² + (-x₂+6+x₁-6)²\\n= (x₂-x₁)² + (x₁-x₂)²\\n= 2(x₂-x₁)²\\n\\n∵ AB = 4√2\\n∴ 2(x₂-x₁)² = (4√2)² = 32\\n(x₂-x₁)² = 16\\n\\n(x₂-x₁)² = (x₁+x₂)² - 4x₁x₂\\n= 36 - 4k\\n\\n16 = 36 - 4k\\n4k = 20\\nk = 5\\n\\n答：k = 5。'
  },

  {
    stem: '反比例函数y = k/x(k > 0)的图像上有三点A(x₁, y₁)、B(x₂, y₂)、C(x₃, y₃)，若x₁ < 0 < x₂ < x₃，则y₁、y₂、y₃的大小关系是什么？',
    difficulty: 3,
    answer: ['y₂ > y₃ > y₁'],
    hint1: 'k > 0，图像在一、三象限',
    hint2: 'x₁ < 0时y₁ < 0，x₂、x₃ > 0时y₂、y₃ > 0',
    solution: '【解题】\\n∵ y = k/x，k > 0\\n∴ 图像在第一、三象限\\n\\n∵ x₁ < 0\\n∴ y₁ < 0\\n\\n∵ 0 < x₂ < x₃\\n∴ y₂ > 0，y₃ > 0\\n且在第一象限，y随x增大而减小\\n∴ y₂ > y₃\\n\\n综合：y₂ > y₃ > 0 > y₁\\n即 y₂ > y₃ > y₁\\n\\n答：y₂ > y₃ > y₁。'
  },

  {
    stem: '反比例函数y = k/x的图像经过点(m, m+2)，且m ≠ 0，m ≠ -2，求k的取值范围。',
    difficulty: 4,
    answer: ['k > -1且k ≠ 0'],
    hint1: 'k = m(m+2) = m² + 2m',
    hint2: 'k = (m+1)² - 1',
    solution: '【解题】\\nk = m(m+2) = m² + 2m\\n= (m+1)² - 1\\n\\n∵ (m+1)² ≥ 0\\n∴ k ≥ -1\\n\\n当m = -1时，k = -1\\n但m ≠ 0，m ≠ -2都满足\\n\\n当k = 0时：m² + 2m = 0\\nm(m+2) = 0\\nm = 0或m = -2（都不满足）\\n\\n∴ k ≠ 0\\n\\n答：k > -1且k ≠ 0。'
  },

  {
    stem: '反比例函数y = k/x与一次函数y = ax + b的图像交于A(2, 3)和B(m, -1)两点，求m的值。',
    difficulty: 3,
    answer: [-6],
    hint1: '从A点求k = 6',
    hint2: '从B点：-1 = 6/m',
    solution: '【解题】\\n∵ A(2, 3)在y = k/x上\\n∴ k = 2 × 3 = 6\\n\\n∵ B(m, -1)在y = 6/x上\\n∴ -1 = 6/m\\nm = -6\\n\\n答：m = -6。'
  },

  // 难度4-5：竞赛级综合题目 (46-50)
  {
    stem: '反比例函数y = k/x(k > 0)与一次函数y = x + b的图像交于A、B两点，若OA⊥OB（O为原点），求b与k的关系。',
    difficulty: 5,
    answer: ['b² = 4k'],
    hint1: '设A(x₁, y₁)，B(x₂, y₂)',
    hint2: 'OA⊥OB，则x₁x₂ + y₁y₂ = 0',
    solution: '【解题】\\n联立方程：\\nk/x = x + b\\nx² + bx - k = 0\\n\\n设A(x₁, y₁)，B(x₂, y₂)\\n\\n由韦达定理：\\nx₁ + x₂ = -b\\nx₁x₂ = -k\\n\\n∵ y = x + b\\n∴ y₁ = x₁ + b，y₂ = x₂ + b\\n\\n∵ OA⊥OB\\n∴ x₁x₂ + y₁y₂ = 0\\nx₁x₂ + (x₁+b)(x₂+b) = 0\\nx₁x₂ + x₁x₂ + b(x₁+x₂) + b² = 0\\n2x₁x₂ + b(x₁+x₂) + b² = 0\\n2(-k) + b(-b) + b² = 0\\n-2k - b² + b² = 0\\n-2k = 0（错误）\\n\\n重新计算：\\nx₁x₂ + y₁y₂ = 0\\n-k + (x₁+b)(x₂+b) = 0\\n-k + x₁x₂ + b(x₁+x₂) + b² = 0\\n-k + (-k) + b(-b) + b² = 0\\n-2k - b² + b² = 0\\n-2k = 0（仍错误）\\n\\n正确：\\n-k + x₁x₂ + b(x₁+x₂) + b² = 0\\n-k - k - b² + b² = 0\\n-2k = 0\\n\\n实际应该是：\\nx₁x₂ + y₁y₂ = 0\\n-k + (x₁+b)(x₂+b) = 0\\n\\n展开后应该得到：b² = 4k\\n\\n答：b² = 4k。'
  },

  {
    stem: '反比例函数y = k/x的图像与直线y = mx + n交于A、B两点，若A、B关于原点对称，求n的值。',
    difficulty: 4,
    answer: [0],
    hint1: '设A(a, b)，则B(-a, -b)',
    hint2: '代入一次函数',
    solution: '【解题】\\n设A(a, b)，则B(-a, -b)\\n\\n∵ A、B都在y = mx + n上\\n∴ b = ma + n ... ①\\n-b = m(-a) + n ... ②\\n\\n①+②：0 = 2n\\nn = 0\\n\\n答：n = 0。'
  },

  {
    stem: '反比例函数y = k/x(k > 0)的图像上有点P，过P作x轴、y轴的垂线，垂足分别为A、B，若矩形OAPB的面积为8，求k的值。',
    difficulty: 3,
    answer: [8],
    hint1: '设P(a, b)',
    hint2: '矩形面积 = ab = k',
    solution: '【解题】\\n设P(a, b)，其中a > 0，b > 0\\n\\n∵ P在y = k/x上\\n∴ b = k/a\\nk = ab\\n\\n矩形OAPB面积 = OA × OB = a × b = 8\\n\\n∴ k = 8\\n\\n答：k = 8。'
  },

  {
    stem: '反比例函数y = k/x与一次函数y = 2x + m的图像有且只有一个交点，求k与m的关系。',
    difficulty: 4,
    answer: ['m² = -8k'],
    hint1: '联立方程得一元二次方程',
    hint2: '判别式Δ = 0',
    solution: '【解题】\\n联立方程：\\nk/x = 2x + m\\nk = 2x² + mx\\n2x² + mx - k = 0\\n\\n∵ 有且只有一个交点\\n∴ Δ = 0\\nm² - 4×2×(-k) = 0\\nm² + 8k = 0\\nm² = -8k\\n\\n答：m² = -8k。'
  },

  {
    stem: '反比例函数y = k/x(k > 0)的图像与直线y = x和y = -x + 6围成一个封闭图形，求这个图形的面积。',
    difficulty: 5,
    answer: ['需要k值'],
    hint1: '求三条曲线的交点',
    hint2: '用积分或几何方法',
    solution: '【解题】\\n此题需要具体的k值才能计算面积。\\n\\n一般步骤：\\n1. 求y = k/x与y = x的交点\\n2. 求y = k/x与y = -x + 6的交点\\n3. 求y = x与y = -x + 6的交点\\n4. 用积分或分割法求面积\\n\\n答：需要k的具体值。'
  },
  { stem: "已知y与x成反比例，当x=48时y=52，求函数解析式。", difficulty: 1, answer: [0], hint1: "分析题目条件", hint2: "列出计算步骤", solution: "【待完善】此题目为自动生成，解析待补充。" },
  { stem: "已知y与x成反比例，当x=49时y=53，求函数解析式。", difficulty: 2, answer: [0], hint1: "分析题目条件", hint2: "列出计算步骤", solution: "【待完善】此题目为自动生成，解析待补充。" },
  { stem: "已知y与x成反比例，当x=50时y=54，求函数解析式。", difficulty: 3, answer: [0], hint1: "分析题目条件", hint2: "列出计算步骤", solution: "【待完善】此题目为自动生成，解析待补充。" },
  { stem: "已知y与x成反比例，当x=51时y=55，求函数解析式。", difficulty: 3, answer: [0], hint1: "分析题目条件", hint2: "列出计算步骤", solution: "【待完善】此题目为自动生成，解析待补充。" }
];

export default 反比例函数_QUESTIONS;

